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Riemann's zeta function is a function with many faces, I mean representations. I recently derived this one, bellow, as a continued fraction over prime numbers.

$$ \zeta(s)=1 +\cfrac{\frac{1}{2^{s}}}{1-\frac{1}{2^{s}} -\cfrac{\frac{2^{s}-1}{3^{s}}}{1+\frac{2^{s}-1}{3^{s}} -\cfrac{\frac{3^{s}-1}{5^{s}}}{1+\frac{3^{s}-1}{5^{s}} -\cfrac{\frac{5^{s}-1}{7^{s}}}{1+\frac{5^{s}-1}{7^{s}} -\cfrac{\frac{7^{s}-1}{11^{s}}}{1+\frac{7^{s}-1}{11^{s}} -\ddots}}}}} $$

... and I'd like to know if this is known in the literature and if so I'd appreciate to have references about it.

Thanks.

Neves
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2 Answers2

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The continued fraction representation above had its origins on another problem I was working on sometime ago.

It's based on a very simple way of looking at the Euler's product representation of $\frac{1}{\zeta(s)}$. Interestingly it applies to every infinite product.

And this is as follows

$$ \frac{1}{\zeta(s)}=\left(1-\frac{1}{2^s}\right)-\left(1-\frac{1}{2^s}\right)\frac{1}{3^s}-\left(1-\frac{1}{2^s}\right)\left(1-\frac{1}{3^s}\right)\frac{1}{5^s}-\cdots $$

From here its easy to derive the above continued fraction using Euler's continued fraction formula.

And thats it, It's nice and eventually a new thing.

EDIT

Just to make it clear, note that $$ \begin{align*} \frac{1}{\zeta(s)}&=\left(1-\frac{1}{2^s}\right)\left[\left(1-\frac{1}{3^s}\right)-\left(1-\frac{1}{3^s}\right)\frac{1}{5^s}-\left(1-\frac{1}{3^s}\right)\left(1-\frac{1}{5^s}\right)\frac{1}{7^s}-\cdots\right]\\ &=\left(1-\frac{1}{2^s}\right)\left(1-\frac{1}{3^s}\right)\left[\left(1-\frac{1}{5^s}\right)-\left(1-\frac{1}{5^s}\right)\frac{1}{7^s}-\cdots\right]\\ &\vdots\\ &=\prod_{p\in\mathbb{P}}\left(1-\frac{1}{p^{s}}\right) \end{align*} $$ where $\mathbb{P}$ is the set of the prime numbers.

EDIT

To derive the continued faction just put $\frac{1}{\zeta(s)}$ in the form $$ \frac{1}{\zeta(s)}=1-\frac{1}{2^s}\left(1+\frac{2^s-1}{3^s}\left(1+\frac{3^s-1}{5^s}\left(1+\frac{5^s-1}{7^s}\left(1+\frac{7^s-1}{11^s}\left(1+\ddots\right ) \right ) \right ) \right ) \right) $$ and then just apply the Euler continued fraction formula. So we can write this as $$ \frac{1}{\zeta(s)}=1-\frac{1}{2^s}-\frac{1}{2^s}\frac{2^s-1}{3^s}-\frac{1}{2^s}\frac{2^s-1}{3^s}\frac{3^s-1}{5^s}-\frac{1}{2^s}\frac{2^s-1}{3^s}\frac{3^s-1}{5^s}\frac{5^s-1}{7^s}-\cdots $$ Now, let $a_1=-\frac{1}{2^s};a_2=\frac{2^s-1}{3^s};a_3=\frac{3^s-1}{5^s};a_4=\frac{5^s-1}{7^s}\cdots$ and we'll get the Euler continued fraction formula.

Neves
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3

By using Mathematica to simplify the first 7 primes, we get: $$\frac{510510^s}{\left(2^s-1\right) \left(3^s-1\right) \left(5^s-1\right) \left(7^s-1\right) \left(11^s-1\right) \left(13^s-1\right) \left(17^s-1\right)},$$ which is equivalent to: $$\prod _{p\text{ prime}} \frac{p^s}{p^s-1} = \zeta(s).$$ Product does not converge when $s=1.$

Scroll down to Euler product formula (2nd paragraph).
When $s=1\text{, }\frac{1}{1-\frac{1}{p^s}}$ simplifies to $\frac{p}{p-1}.$ When $s>1,$ there is no simplification.

Neves's formula puts the exponents back onto the primes when it is simplified, $\frac{p^s}{p^s-1}.$

Fred Kline
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